Hi,

I have this problem that i don't know how to start with:

If E is a set on the real line with lebesgue outermeasure zero then its complement is dense in R

Thank you

Printable View

- Oct 9th 2010, 09:08 AMraa91lebesgue outermeasure
Hi,

I have this problem that i don't know how to start with:

If E is a set on the real line with lebesgue outermeasure zero then its complement is dense in R

Thank you - Oct 9th 2010, 06:50 PMTinyboss
Try the contrapositive: if the complement of E is NOT dense in R, then what can you say about E?

- Oct 10th 2010, 07:58 AMraa91
if it's not dense then its closure is a subset of R and from there we prove that E has a measure different than zero?

- Oct 10th 2010, 08:51 PMTinyboss
If the complement of E is not dense, then E contains an open set.

- Oct 11th 2010, 09:56 AMraa91
ok so if i go roughly like this would it be true:

Suppose E^c is not dense then there exist an interval I in R such that I is contained in E^c.

the lebesque outermeasure is denoterd by ð*. so ð*(E)=ð*(EI)+ð*(EI^c)= 0 + ð*(I^c) which is differenty than zero since I is an interval.